M.Sc Thesis
M.Sc Student Kemarsky Alexander Weil Sums, Tate Thesis and the Semigroup Department of Mathematics Professor Shai Haran

Abstract

We give a q- analogue interpolating between the p-adic and real part of Tate thesis (the local unramified  part ). There is a natural semi-group acting, and we give its p-adic, real and q- analogue versions.

We define the global semi-group by taking tensor product over all primes (finite or real ), and then average over  Q*- orbits . The global semi-group is an integral operator. We give an equivalent formulation of the Riemann Hypothesis in terms of the asymptotic of the kernel of this operator near the origin.

We show the global semi-group is related to the classical semi-group of fractional integration of Riemann-Liouville via summation over the integers and its inverse operation using summation with a Mobius function.

We prove that the global semi-group can be extended to a group on the "algebraic Schwartz space" which consists of generalized theta-functions.

We present a formal connection of the infinitesimal generator of the global semi-group to Weil’s explicit sums. That is we show the explicit sums are given as a normalized trace of an operator depending on a parameter. For certain values of the parameters the operator is the infinitesimal generator of the global semi-group on an appropriate Hilbert space .